Pole Order Spectral Sequence

• Pole Order Spectral Sequence is a homological tool connecting algebraic, topological, and Hodge-theoretic invariants to analyze hypersurface complements, Milnor fibers, and hyperplane arrangements.
• It constructs a spectral sequence by filtering the de Rham complex through controlled pole orders and relates Koszul cohomology to the syzygies among a polynomial's partial derivatives.
• Its applications include computing Hodge and pole order filtrations, determining monodromy, and identifying Bernstein–Sato roots, with degeneration behavior sensitive to singularity types.

The pole order spectral sequence is a homological tool that realizes the interplay between algebraic, topological, and Hodge-theoretic invariants of hypersurface complements, Milnor fibers, and hyperplane arrangements. It provides a deep connection between the cohomological behavior of forms with controlled pole order, the structure of the Koszul complex associated to a defining polynomial, and the syzygies among the polynomial's partial derivatives. Its applications encompass explicit computations of Hodge and pole order filtrations, monodromy, and the roots of Bernstein–Sato polynomials.

1. Foundations and Definitions

Let $S = \mathbb{C}[x_0,\dots,x_n]$ be the homogeneous coordinate ring of %%%%1%%%%; for a homogeneous $f \in S$ of degree $N$, define the hypersurface $D = \{ f=0 \} \subset \mathbb{P}^n$ and its complement $U = \mathbb{P}^n \setminus D$. The pole order filtration on the algebraic de Rham complex is induced by allowing forms with poles of controlled order along $D$, explicitly,

$$P^p(j_* \Omega_U^*) = j_*\Omega_{\mathbb{P}^n}^*((p-s+1)D), \quad (p \geq s)$$

where $j: U \hookrightarrow \mathbb{P}^n$ is the inclusion map. The induced filtration on the cohomology is

$$P^pH^m(U) = \mathrm{Im} \left( H^m(\mathbb{P}^n, P^p j_* \Omega_U^*) \to H^m(\mathbb{P}^n, j_* \Omega_U^*) \right).$$

The Koszul complex $K^*(f)$ on the tuple of partial derivatives $f_i = \frac{\partial f}{\partial x_i}$ is central: $$K^*(f): 0 \to \wedge^{n+1}S^{n+1}(-(n+1)N) \xrightarrow{df \wedge} \cdots \xrightarrow{df \wedge} S \to 0$$ with cohomology groups $H^m(K^*(f))_t$ denoting the graded piece of degree $t$.

2. Construction and Structure of the Spectral Sequence

The pole order filtration induces a spectral sequence

$E_r^{p,q}(U) \implies \mathrm{gr}_P^p H^{p+q}(U)$

whose first page can be identified via the Koszul complex with

$E_1^{p,q}(f) \simeq H^{p+q+1}(K^*(f))_{(q+1)N}.$

Differentials have the form

$d_r: E_r^{p,q} \to E_r^{p+r,\,q-r+1}$

where on $E_1$ the differentials are given by the exterior derivative on polynomial forms, intertwined with the wedge operation $df \wedge$. This set-up extends naturally to the filtered Gauss–Manin complex in the affine or Milnor-fiber context.

For affine space $X = \mathbb{C}^n$ and $D = \{ f = 0 \}$, the construction applies to the algebraic de Rham complex with localization $\Omega_X^k[1/f]$ and pole order filtration $P^pA^k = f^{-p}\Omega_X^k$, producing a similar spectral sequence computing local cohomology supported at the origin.

3. Degeneration Phenomena and Criteria

The degeneration behavior of the spectral sequence is highly sensitive to the singularities of $D$:

• If all singularities are isolated and weighted-homogeneous (e.g., nodal curves or surfaces), then the sequence degenerates at $E_2$: $E_2^{p,q} = E_\infty^{p,q}$ (Dimca et al., 2011).
• For arrangements of hyperplanes in four variables, all third differentials vanish ($d_3 \equiv 0$), resulting in degeneration at $E_3$, and all second differentials vanish outside a finite range ("almost $E_2$-degeneration") (Saito, 2019).
• For central hyperplane arrangements or free, locally quasi-homogeneous hypersurfaces, evidence and conjecture indicate collapse at $E_2$ (Dimca et al., 2017).

$\text{Degeneration Summary}$

Hypersurface type	Degeneration page
Isolated weighted-homogeneous sings.	$E_2$
Central hyperplane arr. ($n\leq 4$)	$E_2$ (almost, then full at $E_3$)
Free, locally qh. divisors	$E_2$

The precise location and nature of non-trivial differentials are determined via Castelnuovo–Mumford regularity bounds on the module of logarithmic derivations (Saito, 2019).

4. Algebraic and Cohomological Consequences

The pole order spectral sequence bridges the Milnor (Jacobian) algebra $M(f) = S/J_f$, the cohomology of $K^*(f)$, and the pole order filtration on the cohomology of $U$ (Dimca et al., 2011). The graded pieces $\mathrm{gr}_P^p H^m(U)$ are governed by the Koszul cohomology. For nodal hypersurfaces, the dimensions of relevant Koszul cohomology groups are encoded by the defect of the corresponding system of hypersurfaces passing through the set of nodes.

For a nodal hypersurface in $\mathbb{P}^3$ of degree $N$, the following formulae hold: $$\begin{align*} \dim\,\mathrm{gr}_P^3 H^3(U) & = p_g(S) = \binom{N-1}{3},\ \dim\,\mathrm{gr}_P^2 H^3(U) & = h^{1,1}(S_s) - 1 - \mathrm{def}_{N-4}(\mathcal{N}),\ \dim\,\mathrm{gr}_P^1 H^3(U) & = 0 \end{align*}$$ where $S_s$ is a smooth surface of degree $N$, and $\mathcal{N}$ is the set of nodes (Dimca et al., 2011).

For Milnor fibers, after Fourier-decomposition under the algebraic monodromy $h$, the pole-order filtration on $H^j(F, \mathbb{C})_\lambda$ is canonically induced from the spectral sequence, and the $E_1$-page relates to Koszul cohomology (Dimca et al., 2017).

5. Relation to Syzygies and Defect Theory

The jumps in the pole order filtration are controlled by syzygies among the partial derivatives of $f$. For nodal hypersurfaces,

$$\dim H^n(K^*(f))_{nN-n-1-k} = \mathrm{def}_k(\mathcal{N})$$

where $\mathrm{def}_k(\mathcal{N})$ is the defect of the linear system of degree-$k$ hypersurfaces through the nodes. Thus, the spectrum and structure of the pole order spectral sequence encode detailed syzygetic information (Dimca et al., 2011).

The computational approach for Milnor fiber monodromy leverages an explicit description of the Jacobian syzygy module. The relevant linear ranks yield dimensions of $E_2$-page entries, enabling explicit calculation of pole order spectra (Dimca et al., 2017).

6. Applications to Bernstein–Sato Roots and Monodromy

The pole order filtration determines the roots of the Bernstein–Sato polynomial supported at the origin when the spectral sequence degenerates appropriately. For central hyperplane arrangements in four variables, all such roots must lie in

$$\left\{ \frac{k}{d} \mid n \leq k \leq 2d-2 \right\}$$

and their multiplicities can be read off from the Hilbert series of the Milnor algebra. The filtration pinpoints which graded pieces $\mathrm{gr}_P^p H^{p+q}$ correspond to which roots, with direct applications to stating and verifying the symmetry of spectra, especially for reflection groups (Saito, 2019, Dimca et al., 2017). For free divisors, the set of roots exhibits symmetry under $\alpha \mapsto 2 - \alpha$.

7. Explicit Examples and Computability

The explicit computation and palindromic symmetry of the pole order spectrum are exemplified as follows (Dimca et al., 2017):

• Braid arrangement $A_4$ in $\mathbb{P}^3$: The spectrum $\mathrm{Sp}^1(f)$ is manifestly symmetric about $\alpha \mapsto 1-\alpha$.
• Coxeter $D_4$-arrangement: The spectrum again displays symmetry about $1/2$.
• Complex reflection arrangement $A(3,3,4)$: The spectrum is perfectly palindromic.
• Non-reflection, non-free arrangement: Symmetry may fail, but computation remains effective.
• Plane curves ($n=2$): The multiplicity $m(\lambda)$ of eigenvalue $\lambda$ is controlled solely by the $E_2$-page.

The computational approach exploits properties of the syzygy module, degrees of generators, and explicit rank calculations, achieving tractability in generic, free, and quasi-homogeneous cases.

• (Dimca et al., 2011) Koszul complexes and pole order filtrations (Dimca, Sticlaru)
• (Saito, 2019) Degeneration of pole order spectral sequences for hyperplane arrangements of 4 variables (Saito)
• (Dimca et al., 2017) Computing Milnor fiber monodromy for some projective hypersurfaces (Dimca, Sticlaru)